a nautilus shell exhibiting the golden ratio

The Golden Ratio and the Fibonacci Sequence

The Number Crunch | 0 comments

I’m here today to prove to you that math can be cool. And applicable to the real world! Now, I’m sure you already knew that. All the same, this is one of the parts of math they might not have taught you about in school. The Fibonacci sequence, sure. But the golden ratio? Let’s go on a ride. We’ll start back in ancient Greece… 

… with this guy named Pythagoras.

Pythagoras was a philosopher, scientist, and mathematician, living from c. 570 to c. 495 B.C. He was from the island of Samos, off the coast of what’s now Turkey. Back then, those islands as well as the western edge of Turkey constituted Ionia, and at that time in Greek history Ionia was one of the intellectual centers of the Greek world. Most of the early, “pre-Socratic” philosophers had their roots in Ionia. These were the guys (and gals) who hypothesized about the nature of the cosmos, the Earth, and human reasoning. They had some pretty nifty ideas. One theorized the atom millennia before it became possible to prove its existence. Others determined that the Earth was, in fact, round.

That’s not to say that all their ideas were correct, though. It’s just to say that they had great imaginations! Practically all of their theories were conceptualized as thought experiments, since they lacked the tools to experiment. In fact, Greek science and mathematics relied solely on thought, not on experimentation, which is why it leans towards the philosophical side of things.

Here’s how Pythagoras’s name got connected to the golden ratio.

One day, Pythagoras was walking along when he heard ringing sounds of different pitches coming out of the blacksmith’s shop. Intrigued by the connection between the sounds — which seemed to come in regular intervals — he went in and found out that the sounds were made by the blacksmith’s hammers. He noticed that the hammers had simple size ratios. One was twice as big as another, another was one and a half times bigger, and so on and so forth. 

Using what he learned, Pythagoras developed a theory about the musical scale, essentially applying the golden ratio over it to explain different musical intervals. The story can’t be proven, so no one really knows whether or not Pythagoras did this, but the man did have a keen interest in music. So it’s possible!

But back to that golden ratio thing. What is the golden ratio, anyway? 

 

line segments demonstrating the golden ratio

First off, what’s a ratio?

A ratio is the relation between two things, let’s call them apples and oranges, showing how many of one thing there are compared to the other. If we had 8 apples and 6 oranges, we would say that the ratio of apples to oranges was 8 : 6. And the ratio of total fruit to apples would be 14 :  8. (I stole this example from Google, by the way.)

When you perform division on the two sides of a ratio, the resulting number can also be called the ratio. As in, the ratio between 3 and 4 is 3 : 4 which is 3/4, which is 0.75. Get it?

The golden ratio is a special ratio. Duh, it’s called golden.

Let’s say you have two numbers forming a ratio. We’ll call the first number and the second number b, and let’s say that is bigger than b. There’s a certain ratio, a : b or a/b. We can make another ratio, a + b : a. It’s kind of like my example with the apples and oranges where I added up total fruit! 

Here’s the important part. If the ratio a : b is equal to the ratio a + b : a, then the numbers and are said to be in the golden ratio. 

Now, there’s a way to find the golden ratio, which is a single number — although it’s not a number we can pin down, because it’s a number whose decimal places go on forever. Like Pi, which you probably did learn about in school. It is about equal to 1.61803.

Now, we’ve gotten to the interesting part.

Do you know about the Fibonacci sequence? It’s a sequence of numbers created when you add up the last two entries to make the next entry. So, if we start at 1 (imagining a 0 at the very start), we get this sequence:

1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89…

Do you see how I obtained the next number in the sequence? I got 2 from adding 1+1 and 3 from adding 2+1 and so on and so forth. It’s pretty simple! You too could come up with more numbers in the sequence. (If you want to, drop a couple in the comments!) 

Now, let’s do something crazy and take the RATIOS between the numbers in the Fibonacci sequence. 

1/1 = 1 and 2/1 = 2 and 3/2 = 1.5 and 5/3 = 1.66667 (approx.) and 8/5 = 1/6. 

I could keep going, but let’s discuss the point now…

 

The image above shows a spiral drawn through golden rectangles, which are rectangles laid out in the golden ratio. Notice that the numbers also correspond to the Fibonacci sequence?

What you might have noticed when we took the ratios between the numbers of the Fibonacci sequence was that they approached the number I gave you for the golden ratio. That is, they kept getting closer and closer to it each time. At first, the ratio was way off. 1 is not very close to 1.61803. But by the time we got to 8/5, we had 1.6, which is pretty darn close! Only a hundredth and a little bit off. And from there, the ratio gets closer and closer. 

So there’s a special relationship between the golden ratio and the Fibonacci sequence. But what really makes the golden ratio cool? 

Well, the fact that it’s found pretty much everywhere in the natural world, as it turns out.

 

examples of the golden ratio in nature

And it’s found in art, because it turns out the art and architecture arranged according to the golden ratio is extremely pleasing to the human eye. You might say that very good artists are very attuned to the golden ratio! Here are a couple famous examples…

The golden ratio in art
The golden ratio in art

Isn’t it interesting how the golden ratio is found in nature and in human creations? Sometimes, artists include it purposefully. Other times, it comes naturally. Over the centuries, some have gone as far as to say it’s proof of some Creator with some greater plan for the universe. I wouldn’t go quite that far — although depending on your beliefs you might agree — but the golden ratio is certainly a fascinating concept that deserves thought and attention.

In conclusion

I hope you’ve enjoyed this first entry into “The Number Crunch” category! Math is a great love of mine, and I’ve been wanting to share some cool mathematical concepts with people for a long time. If you enjoyed this article to the max, please pass it along using these handy share buttons! You can also follow Voyage of the Mind using the buttons in the sidebar at the top of this page, or enter your email at the very bottom to join our mailing list. And if you’re loving our work, please consider supporting us on Ko-fi

 

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